Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2409.18045

Necessary and sufficient conditions for universality limits

Benjamin Eichinger·Milivoje Lukić·Harald Woracek
View original source ↗Read source document ↗Export metadata ↓

Abstract

We derive necessary and sufficient conditions for universality limits for orthogonal polynomials on the real line and related systems. One of our results is that the Christoffel-Darboux kernel has sine kernel asymptotics at a point $\xi$, with regularly varying scaling, if and only if the orthogonality measure (spectral measure) has a unique tangent measure at $\xi$ and that tangent measure is the Lebesgue measure. This includes all prior results with absolutely continuous or singular measures. Our work is not limited to bulk universality; we show that the Christoffel-Darboux kernel has a regularly varying scaling limit with a nontrivial limit kernel if and only if the orthogonality measure has a unique tangent measure at $\xi$ and that tangent measure is not a point mass. The possible limit kernels correspond to homogeneous de Branges spaces; in particular, this equivalence completely characterizes several prominent universality classes such as hard edge universality, Fisher-Hartwig singularities, and jump discontinuities in the weights. The main part of the proof is the derivation of a new homeomorphism. In order to directly apply to the Christoffel-Darboux kernel, this homeomorphism is between measures and chains of de Branges spaces, not between Weyl functions and Hamiltonians. In order to handle limits with power law weights, this homeomorphism goes beyond the more common setting of Poisson-finite measures, and allows arbitrary power bounded measures.

Explore related subjects

math.CAmath-phmath.CVmath.MPmath.SP

Keep this discovery

BibTeXRIS

Benjamin Eichinger, Milivoje Lukić, Harald Woracek. 2024-09-26. Necessary and sufficient conditions for universality limits. https://arxiv.org/abs/2409.18045

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the log-concavity of the composite Bessel function $x^{\alpha}J_{\nu }\left( \beta x^{\gamma}\right) $

For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.$ It is well known that the positivity of $v\left( f\right) $ implies that the function $\left\vert f\right\vert $ is strictly log-concave on each subinterval which does not contain zeros of $f.$ In this paper we provide criteria for the positivity of $v\left( F\right) $ for the composite Bessel function $F\left( x\right) =J_{\alpha,\beta,\gamma,\nu}\left( x\right) :=x^{\alpha}J_{\nu}\left( \beta x^{\gamma}\right) $ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu.$

math.CA↗
arXiv2026

Riesz capacity ratios with negative exponents

We investigate sharp inequalities for ratios of Riesz capacities with negative exponents by combining computational experiments with rigorous analysis. For finite subsets of the line, we prove positivity of equilibrium masses when $-1<p<0$, enabling numerical tests of conjectured extremal ratios. In the plane, comparisons of the disk with regular polygon vertex sets reveal a cascade of transitions among the tested competitors and suggest a precise conjecture for the equilibrium measure of odd polygons, for which we give a partial proof. Numerical intersections of equality curves show that the regions where these sets outperform the disk are not simply nested. Similar numerical intersections occur in three dimensions between the regular-simplex equality curve and those of explicit five-point and six-point configurations. Motivated by the dimensional dependence of these comparisons, we prove that for each fixed $p<-2<q<0$, the regular simplex has a larger capacity ratio than the ball in all sufficiently large dimensions. Accompanying Python and Mathematica code supports reproduction and further testing of the conjectures.

math.CA↗
arXiv2026

Shorter proof of dimension-free $L^p$ estimates for maximal Riesz transforms

We provide a shorter and more direct proof of $L^p$ estimates for maximal Riesz transforms (of an arbitrary order) in terms of the corresponding Riesz transforms, with a constant independent of the dimension of the Euclidean space $\mathbb R^d$. This result was originally proved by Mateu, Orobitg, P\'erez and Verdera with a constant depending on the dimension, and improved to a dimension-free inequality by Kucharski, Wr\'obel and Zienkiewicz.

math.CA↗